Representation of Bounded Convex Sets By Rational Convex Hull Of Its Gamma-Extreme Points

H. Minkowski proved C= convext C whenever C is a compact convex subset of a finite-dimensional linear space. If C is bounded but not closed, this representation does not hold anymore. In this case, we introduce the set of so-called γ-extreme points extγC of C and show C = convext γ C = raco extγ C, where raco M denotes the rational convex hull of M.     

Basic Module Theory for General Rings

We show how to construct a theory of modules over general rings (i.e., rings which do not necessarily have an identity element) in such a form that most of the known results of the usual theory of modules over unital rings are obtained by particularizing the results of the general theory.     

Two Open Questions On Goldie Extending Modules

Five open questions on Goldie extending modules were posed by Akalan et al. [1 Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The first one and the second one are considered in this article. It is shown that a 𝒢-extending module M with (C ₃) is 𝒢⁺-extending. Moreover, let M = M ₁ ⊕ M ₂ be a direct sum of 𝒢-extending modules, where M satisfies (C ₃) and M ₁ is M ₂-ojective (or M ₂ is M ₁-ojective), then M is 𝒢-extending. Other sufficient conditions for a direct sum of 𝒢-extending modules to be 𝒢-extending are obtained.     

Cone Length for DG Modules and Global Dimension of DG Algebras

As the definition of free class of differential modules over a commutative ring in [1 Avramov , L. L. , Buchweitz , R.-O. , Iyengar , S. ( 2007 ). Class and rank of differential modules . Invent. Math. 169 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A ^op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D ^c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A ^e -module A.     

Rings Characterized by a Class of Modules

ABSTRACT

Let ? be a class of right R-modules. The notions of ?-injectivity and ?-flatness are used to investigate right ?-Noetherian rings, right ?-hereditary rings, and right ?-coherent rings. For an almost excellent extension S ≥ R, if either ring is the above-mentioned ring, so is the other. Specializing the class ?, some known results can be obtained and extended as corollaries.     

收敛幂级数环的一个性质

本文证明了收敛幂级数环的张量积不是Ｎｏｅｔｈｅｒ环，并提出一个问题。